Euler’s Formula and Platonic solids |
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Five Platonic Solids A
polyhedron is simply a three-dimensional solid which consists of a collection
of polygons, usually joined at their edges. The Platonic solids are convex regular
polyhedra. Each one has identical regular faces, and identical regular vertex
figures. There are only five Platonic Solids. |
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Tetrahedron Hexahedron Octahedron Dodecahedron Icosahedron (Right pyramid)
(Cube) Thanks Rudiger Appel for
his animations: http://www.3quarks.com/GIF-Animations/PlatonicSolids/
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Formulas discovery Let m be the number of polygons meeting at a vertex, v be the number of vertices of each polygon (on one face), F be the number of Faces of the polyhedron, E
be the number of Edges of the polyhedron, and V be the number of Vertices of the polyhedron. Fill in the following
table and discover some formulas from the table. (You can
highlight the cells using a mouse to see the answers!!)
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Formulas 1. 2E =
mF = vV 2. F + V
– E = 2 |
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Euler’s Formula The
formula F + V – E = 2
is called the Euler’s Formula. The formula is also true for any
polyhedra, not just for Platonic’s solids. |
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Exercise 1. Verify that
the Euler’s formula is good for other solids, such as pyramid with a square
base and prism with a pentagon base. Draw
diagrams to help you to check. 2. The Platonic
solids have an interesting dual relationships. To make the dual of a
solid, place a vertex in the center of each of the solid's faces. Then
connect each vertex to the vertices on the adjacent faces. For each Platonic
solid, the result is another Platonic solid. (see the diagram on the right
for the dual of a cube) What are the duals of all five Platonic solids? 3. Make some
interesting paper models using the link: It
is a good project for holidays. 4. Go to the
link with Java Applet: http://www.shef.ac.uk/~pm1nps/courses/groups/plato.html You
can turn the polyhedra round to investigate. Have fun! |
Highlight
the cells for answers:
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